Line Resonance Analysis System

ABSTRACT

The present invention provides a method for monitoring a condition of an electrical cable, the method comprising providing a reference signal CH 0  and a signal CH 1 , said signal CH 1  being the reference signal CH 0  after amplitude and phase modulation by a cable impedance Z DUT  of the electrical cable; calculating the cable impedance Z DUT  as a function of the applied signal frequency based on the reference signal CH 0  and the signal CH 1 ; and analyzing said cable impedance providing an assessment of the cable condition. A system performing the method is also disclosed.

The present invention provides a system and a method for monitoring ofinstalled electrical cables based on line resonance analysis. Monitoringcomprises e.g. condition monitoring and real-time diagnosis of theelectric cables. Throughout the present description the system is alsoreferred to as LIRA (Line Resonance Analysis System). The cables havelengths ranging from a few meters to several hundred kilometers,depending on the cable structure and attenuation.

BACKGROUND

LIRA (Line Resonance Analysis System) is based on transmission linetheory, an established and well documented theory that is at the base oftwo other existing cable fail detection techniques known as “Time DomainReflectometry” (TDR) and “Frequency Domain Reflectometry” (FDR).Differences and limitations of these two existing techniques areexplained in the following.

A transmission line is the part of an electrical circuit providing alink between a generator and a load. The behavior of a transmission linedepends by its length in comparison with the wavelength λ of theelectrical signal traveling into it. The wavelength is defined as:

λ=v/f  (1)

where v is the speed of the electric signal in the wire (also called thephase velocity) and f the frequency of the signal.

When the transmission line length is much lower than the wavelength, asit happens when the cable is short (i.e. few meters) and the signalfrequency is low (i.e. few KHz), the line has no influence on thecircuit behavior. Then the circuit impedance (Z_(in)), as seen from thegenerator side, is equal to the load impedance at any time.

However, if the line length is higher than the signal wavelength, (L≧λ), the line characteristics take an important role and the circuitimpedance seen from the generator does not match the load, except forsome very particular cases.

The voltage V and the current I along the cable are governed by thefollowing differential equations, known as the telephonists equations:

$\begin{matrix}{\frac{^{2}V}{z^{2}} = {\left( {R + {{j\omega}\; L}} \right)\left( {G + {{j\omega}\; C}} \right)V}} & (2) \\{\frac{^{2}I}{z^{2}} = {\left( {R + {{j\omega}\; L}} \right)\left( {G + {{j\omega}\; C}} \right)I}} & (3)\end{matrix}$

where ω is the signal radial frequency, R is the conductor resistance, Lis the inductance, C the capacitance and G the insulation conductivity,all relative to a unit of cable length.

These four parameters completely characterize the behavior of a cablewhen a high frequency signal is passing through it. In transmission linetheory, the line behavior is normally studied as a function of twocomplex parameters. The first is the propagation function

γ=√{square root over ((R+jωL)(G+jωC))}{square root over((R+jωL)(G+jωC))}  (4)

often written as

γ=α+jβ  (5)

where the real part α is the line attenuation constant and the imaginarypart β is the propagation constant, which is also related to the phasevelocity v, radial frequency ω and wavelength λ through:

$\begin{matrix}\begin{matrix}{\beta = \frac{2\pi}{\lambda}} \\{= \frac{\omega}{v}}\end{matrix} & (6)\end{matrix}$

The second parameter is the characteristic impedance

$\begin{matrix}{Z_{0} = \sqrt{\frac{R + {{j\omega}\; L}}{G + {{j\omega}\; C}}}} & (7)\end{matrix}$

Using (4) and (7) and solving the differential equations (2) and (3),the line impedance for a cable at distance d from the end is:

$\begin{matrix}\begin{matrix}{Z_{d} = \frac{V(d)}{I(d)}} \\{= {Z_{0}\frac{1 + \Gamma_{d}}{1 - \Gamma_{d}}}}\end{matrix} & (8)\end{matrix}$

Where ┌_(d) is the Generalized Reflection Coefficient

Γ_(d)=Γ_(L) e ^(−2γd)  (9)

and ┌_(L) is the Load Reflection Coefficient

$\begin{matrix}{\Gamma_{L} = \frac{Z_{L} - Z_{0}}{Z_{L} + Z_{0}}} & (10)\end{matrix}$

In (10) Z_(L) is the impedance of the load connected at the cable end.

From eqs. (8), (9) and (10), it is easy to see that when the loadmatches the characteristic impedance, ┌_(L)=┌_(d)=0 and thenZ_(d)=Z₀=Z_(L) for any length and frequency. In all the other cases, theline impedance is a complex variable governed by eq. (8), which has theshape of the curves in FIG. 1 (amplitude and phase as a function offrequency).

Existing methods based on transmission line theory try to localize localcable failures (no global degradation assessment is possible) by ameasure of V (equation (2)) as a function of time and evaluating thetime delay from the incident wave to the reflected wave. Lineattenuation and environment noise in real environments limit itssensitivity preventing the possibility to detect degradations at anearly stage, especially for cables longer than few kilometers. Inaddition, no global cable condition assessment is possible, which isimportant for cable residual life estimation in harsh environmentapplications (for example nuclear and aerospace applications).

SUMMARY OF THE INVENTION

The object of the invention is to solve or at least alleviate theproblems indicated above.

In a first aspect the invention provides a method of monitoring acondition of an electrical cable, the method comprising providing areference signal CH0, and a signal CH1, said signal CH1 being thereference signal CH0 after amplitude and phase modulation by a cableimpedance Z_(DUT) of the electrical cable; calculating the cableimpedance Z_(DUT) as a function of the applied signal frequency based onthe reference signal CH0 and the signal CH1; and analyzing said cableimpedance providing an assessment of the cable condition.

In an embodiment the reference signal CH0 may be a white noise signalhaving a frequency bandwidth from w₁ to w₂, or a frequency sweep from afrequency w₁ to a frequency w₂, w₁ and w₂ depending on the cable length.The method may further comprise calculating an amplitude and phase ofthe cable impedance Z_(DUT) as a function of frequency, by calculatingthe averaged windowed transfer function from the reference signal CH0 tothe signal CH1 by using

$Z_{DUT} = {Z_{1}\left( {\frac{V_{0}}{V_{1}} - 1} \right)}$

where Z₁ is an internal impedance of a digitizer channel for signal CH1,and V₀ and V₁ are the voltage phasors measured at a digitizer channelfor reference signal CH0 and a digitizer channel 1 for signal CH1,respectively. In an even further embodiment, the method may compriseevaluating at least one resonance frequency pattern of the cableimpedance Z_(DUT), wherein said at least one resonance frequency patternprovides information concerning cable degradation.

Said at least one resonance frequency pattern may be a Fourier transformof the impedance phase and amplitude, the method further comprisingidentifying a fundamental frequency f′ in the Fourier transform due to awave reflection at a distance d, wherein said distance d is a cabletermination, identifying a further frequency component f″ in the Fouriertransform due to a wave reflection at a location x, wherein said wavereflection at location x is due to a discontinuity in electricalparameters of said cable, and calculating said location x by:

$\frac{f^{\prime}}{f^{''}} = \frac{d}{x}$

d being the length of said cable.

Said at least one resonance frequency pattern may be a Fourier transformof the impedance phase and amplitude, the method may in an alternativeembodiment comprise identifying a fundamental frequency f′ in theFourier transform due to a wave reflection at a distance d, wherein saiddistance d is a cable termination, identifying a further frequencycomponent f″ in the Fourier transform due to a wave reflection at alocation x, wherein said wave reflection at location x is due to adiscontinuity in electrical parameters of said cable, and calculatingsaid location x by:

$x = \frac{v_{r}v_{0}f^{''}}{2}$

wherein v_(r) is the relative phase velocity of an electrical signal inthe cable, and v₀ the speed of light in vacuum.

In an even further embodiment of the invention said at least oneresonance frequency pattern is a Fourier transform of the impedancephase and amplitude, the method further comprising identifying twoconsecutive resonance frequency values f_(k) and f_(k+1) respectively,of the cable impedance Z_(DUT); calculating a first value of a relativephase velocity v_(r) of the cable by applyingv_(r)=2L(f_(k+1)−f_(k))/v₀, where L is the length of the cable and v₀ isthe light speed in vacuum; calculating a fundamental frequency f′ ofsaid cable using said first relative phase velocity v_(r) and applying

${f^{\prime} = \frac{2L}{v_{r}v_{0}}},$

where L is the length of the cable and v₀ is the light speed in vacuum;calculating a second value f″ of said fundamental frequency f′ byfinding a maximum peak value in the Fourier transform in the domain oft′ in a selectable interval around f′; and calculating a final relativephase velocity v^(final) _(r) by applying

${v_{r}^{final} = \frac{2\; L}{v_{0}f^{''}}},$

where L is the length of the cable, v₀ is the light speed in vacuum,wherein said relative phase velocity provides information about globaldegradation of said cable.

In an even further embodiment providing information about globaldegradation of the cable, the method comprises calculating a cableimpedance phase shift, wherein said reference signal CH0 is a multi-tonesine wave at a resonance frequency, and calculating a phase differencebetween CH0 and CH1 in time domain.

In a second aspect the invention provides a system of conditionmonitoring of an electrical cable, the system comprising a generatingmeans providing a reference signal CH0; an acquisition means acquiringsaid reference signal CH0 and a signal CH1, said signal CH1 being thereference signal CH0 after amplitude and phase modulation by a cableimpedance Z_(DUT) of the electrical cable; and an analyzing meanscalculating the complex cable impedance Z_(DUT) as a function of theapplied signal frequency based on the reference signal CH0 and thesignal CH1, and analyzing said cable impedance providing an assessmentof cable condition and/or cable failure.

Said generating means may be operative to provide a reference signal CH0selected from a group consisting of a white noise signal having afrequency bandwidth from 0 to a selectable frequency w₂, a sweep signalhaving a frequency bandwidth from w₁ to w₂ (both selectable), and amulti-tone sine wave at selected resonance frequencies.

The acquisition means may be a digital storage oscilloscope.

The system may further comprise a simulator operative to provide theanalyzing means with said reference signal CH0 and said signal CH1 basedon cable parameters, load parameters and transmission line equations. Amodulator may also be arranged between said generating means and saidacquisition means, said modulator being connected to the electricalcable, and being operative to output the reference signal CH0 and saidcable impedance phase and amplitude modulated signal CH1.

In the claims the term monitor is to be widely interpreted and includese.g. global/local condition monitoring, real-time diagnosis, and faultdetection.

LIRA (Line Resonance Analysis System) improves the detection sensitivityand accuracy by analyzing the cable input impedance (equation (8) andFIG. 1). In essence, the following steps are provided:

-   -   Local degradation detection and localization:        -   a. Noiseless estimation of the line input impedance as a            function of frequency (bandwidth 0-X MHz, where X depends on            the cable length).        -   b. Use of a hardware module called “Modulator” for the            estimation of the line input impedance.        -   c. Spectrum analysis of the line input impedance to detect            and localize degradation spots (see detailed explanation).    -   Global degradation assessment:        -   a. Same as point a above        -   b. Same as point b above        -   c. Spectrum analysis of the line input impedance to estimate            the right value of the phase velocity (equation (1)). The            phase velocity is used as a condition indicator of the cable            global state.    -   These steps are explained in detail later in this document.

LIRA (Line Resonance Analysis System) according to the invention is ableto monitor the global, progressive degradation of the cable insulationdue to harsh environment conditions (e.g. high temperature, humidity,radiation) and detect local degradation of the insulation material dueto mechanical effects or local abnormal environment conditions. In thiscase, LIRA can estimate the location of the challenged part with anestimation error within 0.5% of the cable length.

The LIRA system may be used for detecting and monitoring insulationdegradation and line breakage in all kinds of electrical cables (powerand signal cables); i.e. cables in power generation, distribution andtransmission, cables in process industries, cables in aerospaceindustry, on cable in onshore, offshore and subsea installations, andidentify the position of the damage/breakage. The monitoring anddetection may be performed from a remote location in real time.

The invention is defined in the appended claims.

BRIEF DESCRIPTION OF DRAWINGS

Embodiments of the invention will now be described with reference to thefollowing drawings, where:

FIG. 1 shows a graphical representation of a complex line impedanceamplitude and phase as a function of frequency according to equation(8);

FIG. 2 shows hardware and software modules of a system according to anembodiment of the invention;

FIG. 3 shows a functional design of a modulator according to anembodiment of the present invention;

FIG. 4 shows a functional diagram of a spot detection algorithmaccording to an embodiment of the invention;

FIG. 5 is a power spectrum of a phase impedance in a domain of t′, wherethe x-axis has been scaled to distance from cable start (d=300 m),according to an embodiment of the present invention;

FIG. 6 is a power spectrum of phase impedance in the domain of t′, wherethe frequency component at x=200 is visible, due to a capacity change of20 pF in a 30 cm cable segment, according to an embodiment of thepresent invention; and

FIG. 7 illustrates a real-time monitoring of impedance phase shift in ananalyzing means according to an embodiment of the invention.

DETAILED DESCRIPTION

FIG. 2 shows an embodiment of the system with hardware and softwaremodules. These modules will be described below.

-   -   The Arbitrary Wave Generator. It is driven by the LIRA Generator        software to supply the system with a reference signal CH0. The        reference signal can be:        -   A white noise signal.        -   A sweep signal, from 0 Hz to the selected bandwidth. Same            effect as of a white noise signal.        -   A multi-tone sine wave. This is used for real-time            monitoring of impedance phase shifts.    -   The Modulator. A functional diagram of the modulator is shown in        FIG. 4. The output of this module is the reference signal (CH0),        distorted by the generator internal impedance Rg, and a phase        and amplitude modulated signal (CH1), which is modulated by the        frequency dependent impedance Z_(DUT) of the cable provided to        the modulator through a cable connection.    -   The modulator functional diagram is shown in FIG. 4, where DUT        is the connection to the cable under test. The impedance at DUT        is calculated as:

$\begin{matrix}{Z_{DUT} = {Z_{1}\left( {\frac{V_{0}}{V_{1}} - 1} \right)}} & (11)\end{matrix}$

-   -    where Z₁ is the digitizer channel 1 (CH1) internal impedance        (50 ohm) and V₀ and V₁ are the voltage phasors measured at        digitizer channel 0 (CH0) and channel 1 (CH1).    -   From equation (11) it follows that there is no influence on        Z_(DUT) from Rb, Rg and any parasitic impedance on the left of        CH0. Z₁ has some known capacitance (15 pF) that the system can        easily take into consideration.    -   Equation (11) shows that the cable input impedance is a simple        function of the reciprocal of the transfer function between V₀        and V₁, both acquired by the digital storage oscilloscope. LIRA        performs a windowed transfer function with an average technique        to remove noise and applies eq. (11) to estimate the line        impedance in the applicable bandwidth.    -   The 2-channel Digital Storage Oscilloscope. It is used to        acquire CH0 and CH1. This is a commercial hardware unit.    -   The LIRA simulator. This module can be operated stand alone, or        it can be connected to the LIRA Analyzer. In the last case the        LIRA simulator works in frequency domain applying the        transmission line equations (Eqs. 1 to 10) and then performing        an inverse Fourier transformation to provide the analyzer with        the 2 time domain signal channels (CH0, CH1), as they came        directly from the Modulator connected to the tested cable. In        addition to that, it employs a stochastic model to evaluate the        uncertainties in the cable electrical parameters due to        manufacturing tolerances and environment changes. Cable        parameters and load parameters for the actual cable connection        are input to the LIRA simulator.    -   The stochastic model evaluates and applies statistical        variations (using a normal distribution with user selected        standard deviation) in the electrical parameters (L, C and R)        along the cable, due to manufacturing tolerances and environment        noise.    -   The LIRA Analyzer. It can be operated in real or simulation        mode. In the first case it takes the input from the 2-channel        Digital Storage Oscilloscope, in the second case the input comes        from the LIRA simulator module. The LIRA Analyzer is the core of        the wire monitoring system. The LIRA Analyzer works both in        frequency and time domain, performing the following tasks:        -   Estimate and display the frequency spectrum of the line            input impedance.        -   Calculate the resonance frequencies. Resonance frequencies            are calculated from the impedance spectrum and correspond to            frequency values where the phase is zero.        -   Estimate the cable characteristic impedance Z_(DUT). It is            also calculated from the impedance spectrum. The            characteristic impedance is the value of the impedance            amplitude at any local maximum (or minimum) of the impedance            phase.        -   Estimate the cable length, if not known.        -   Detect local degradation areas and localize it.        -   Detect load changes.        -   Measure and display the amplitude ratio and the phase shift            between the 2 acquired channels (CH0 and CH1). This is done            when the reference signal is a multi-tone sine wave and LIRA            evaluates in time domain the phase shift between the two            channels. The phase shift is initially zero, at resonance            conditions, and any deviation from that can be correlated to            a change in the average cable electrical parameters.

Diagnosis and Localisation of Local Degradation

LIRA implements 2 algorithms for the detection of local insulationdefects, referred as the PRN (Pseudo Random Noise) method and the SWEEPmethod. The PRN method is the preferred one for cable lengths below 200m, while the SWEEP method is used for longer cables. Both methods followthe scheme in FIG. 4 and they differ only in the shape of the generatedreference signal CH0. These methods will therefore be explained in thefollowing with reference to the PRN method, but the explanation will beequally applicable for the SWEEP method.

In the PRN method, the line impedance is calculated as the averagedwindowed transfer function from the reference signal CH0 to theimpedance modulated signal CH1, which result in the calculation of theamplitude and phase of the line impedance Z_(DUT) as a function offrequency (See Eq. (11)). Once the line impedance is calculated, thecable state is analyzed by the examination of the frequency content ofthe amplitude and phase components of the complex impedance. This willbe explained in the following.

Eq. (8) is the mathematical expression of the function in FIG. 1.Actually the line impedance Z_(d) (for a cable at distance d from theend of the cable), is a complex parameter and FIG. 1 shows both theamplitude and phase of it. The pseudoperiodic shape of the phase is dueto the periodicity of ┌_(d), Eq. (9), that can be rewritten as:

Γ_(d)=Γ_(L) e ^(−2αd) e ^(−2jβ)  (12)

where the amplitude is decreasing with d (the cable length) because ofthe attenuation α (the phase is periodic if α=0). The period of ┌d (andthen of the line impedance phase) is ½β, considering d as theindependent variable, or ½d, considering β the independent variable (asin FIG. 1).

Using the expression for the propagation constant β from Eq. (6), Eq.(12) can be rewritten as:

$\begin{matrix}{\Gamma_{d} = {\Gamma_{L}^{{- 2}\alpha \; d}^{\frac{{- 2}{j\omega}\; d}{v}}}} & (13) \\{\Gamma_{d} = {\Gamma_{L}^{{- 2}\; \alpha \; d}^{\frac{{- 4}\; {j\pi}\; {fd}}{v}}}} & (14)\end{matrix}$

where f is the frequency of the applied signal and v is the phasevelocity of the electrical signal in the cable.

Assuming f as the independent variable and writing the followingtranslations:

$\begin{matrix}{f->t^{\prime}} & (15) \\{\frac{4\pi \; d}{v_{r}v_{0}}->\omega^{\prime}} & (16)\end{matrix}$

where v_(r)=v/v₀, v_(r) being the relative phase velocity of theelectrical signal in the cable, and v₀ the speed of light in vacuum.

Γ_(L) e ^(−2αd) =A  (17)

Eq. (14) becomes:

Γ_(d) =Ae ^(−jω′t′)  (18)

Equation (18) is the mathematical expression of a pseudo-periodicfunction of radial frequency ω′ and amplitude A. In the lossless case(α=0) A=1, in real life lossy cables α is an increasing function ofsignal frequency, so that amplitude A is a decreasing function of t′,resulting in the damped oscillation of FIG. 1. The frequency of thisfunction (in the domain of t′) is:

$\begin{matrix}\begin{matrix}{f^{\prime} = \frac{\omega^{\prime}}{2\pi}} \\{= \frac{2\; d}{v_{r}v_{0}}}\end{matrix} & (19)\end{matrix}$

where f′ is the fundamental frequency of the phase function in thedomain of t′ due to the wave reflection at distance d (the cabletermination). Note that the expression of f′ has the dimension of timeand it is the time elapsed for a wave to reach the termination at thedistance d and be reflected back. The Fourier transform (power spectrum)of the impedance phase, in the domain of t′, looks e.g. like FIG. 5,where the x-axis has been scaled to d at the fundamental frequency givenin Eq. (19). In FIG. 5, the x-axis has been scaled to distance fromcable start (d=300 m).

When at a distance x the wave finds a discontinuity in the electricalparameters of the cable (for example a small change in the insulationdielectric), another reflection would be visible from distance x, whichwould add a new frequency component in the power spectrum of theimpedance phase, where the frequency (from Eq. (19)) would be:

$\begin{matrix}{f^{''} = \frac{2x}{v_{r}v_{0}}} & (20)\end{matrix}$

And so:

$\begin{matrix}{\frac{f^{\prime}}{f^{''}} = \frac{d}{x}} & (21)\end{matrix}$

If the cable length is known, the knowledge of f′ and f″ from the powerspectrum of the impedance phase (in the domain of t′) is sufficient forthe calculation of the x location:

$\begin{matrix}{x = {d\frac{f^{''}}{f^{\prime}}}} & (22)\end{matrix}$

If d is not known, the knowledge of the relative phase velocity v_(r)(from the cable datasheet or by measuring it on a cable sample of thesame type) can be used to calculate the x location based on Eq. (19):

$\begin{matrix}{x = \frac{v_{r}v_{0}f^{''}}{2}} & (23)\end{matrix}$

The final result is then a spike at any position where a change (even avery small change) of electrical parameters (mainly a dielectric valuechange) produces a reflected wave of the applied reference signal. Thisreflection appears as a frequency component in the phase/amplitudespectrum of the line impedance. The frequency of the reflected wave is alinear function of the distance from the cable end to the deviation.FIG. 6 shows a Fourier transform (power spectrum) of phase impedance inthe domain of t′, where a reflection due to a change of electricalparameters of the cable at location x=200 m from cable start is visibleas a frequency component at x=200. This spike is in the case of FIG. 6,due to a capacity change of 20 pF in a 30 cm segment of the cable undertest.

When condition monitoring a cable, a number of discontinuities (n) inthe electrical parameters of the cable may be present. Each of thesediscontinuities will appear in the power spectrum as distinct frequencycomponents spikes f^(n), and their positions x_(n) identified asexplained above.

In order to have good sensitivity and digital resolution, it isimportant to operate with the highest possible bandwidth, which ishowever limited by the cable attenuation. Successful tests have beenperformed with 30 m cables (100 MHz bandwidth) up to 120 Km cables (20KHz bandwidth) at the time of writing. Long cables require narrowbandwidths to overcome the increasing cable attenuation, which is afunction of frequency.

Global Degradation Monitoring

LIRA makes use of 2 different methods for monitoring global changes inthe cable condition:

-   -   1. The Relative Phase Velocity estimation and monitoring.    -   2. The line impedance phase shift from any resonance condition.

As for the local degradation and diagnosis, the first method for globaldegradation is also based on applying a reference signal CH0 having afrequency bandwidth from w₁ to w₂, which is then phase and amplitudemodulated by the cable impedance Z_(DUT) of the cable under test,providing the signal CH1. The second method is based on applying amulti-tone sine wave as a reference signal CH0. The analysis of theresulting signals CH0 and CH1 is explained in detail for the two methodsbelow.

Method 1:

The Relative Phase Velocity is calculated by LIRA through a 2 stepprocess:

-   -   1. A first approximate value is estimated using 2 consecutive        resonance frequency values in the line impedance, applying the        following equations: At any resonance, the cable length L is        exactly-equal to half wavelength or any multiple of that (this        is true when the cable is open ended, but different load        reactances can be easily accounted for), or, using Eq. (1):

$\begin{matrix}{L = {\frac{v_{r}v_{0}}{2f_{k}}k}} & (24)\end{matrix}$

-   -    where L is the cable length, v₀ is the light speed in vacuum,        v_(r) is the relative phase velocity and f_(k) is the k^(th)        resonance peak frequency.    -    Applying Eq. (24) to two consecutive resonance peaks, we get:

$\begin{matrix}{v_{r} = {2{{L\left( {f_{k + 1} - f_{k}} \right)}/v_{0}}}} & (25)\end{matrix}$

-   -    Eq. (25) is used by LIRA to evaluate a first value of v_(r),        after the estimation of the cable input impedance and the        calculation of the resonance frequencies. Note that any load        reactance shift would be eliminated by the difference term in        the equation. The reason why this value is approximate is that        v_(r) is a slow function off, but it has been assumed constant        in Eq. (25).    -   2. The value of v_(r) found in step 1 is used to calculate the        approximate value of the fundamental frequency f′ (domain of t′,        see description above and FIG. 5), as:

$\begin{matrix}{f^{\prime} = \frac{2L}{v_{r}v_{0}}} & (26)\end{matrix}$

-   -    LIRA searches the maximum peak f″ in the Fourier transform        (power spectrum) in the domain of t′, in a user selectable        interval around f′. When an accurate value of f′ (called f″) is        found from the spectrum, Eq. (19) is solved for v_(r) as:

$\begin{matrix}{v_{r}^{final} = \frac{2L}{v_{0}f^{''}}} & (27)\end{matrix}$

-   -    which is the final and accurate value of the phase velocity.        The phase velocity decreases with the degradation of the cable        insulation.

Method 2:

The line impedance has zero phase at any resonance condition. Using asreference signal a sine wave at a resonance frequency, the 2 outputsignals CH0 and CH1 are in phase. The phase difference between CH0 andCH1 (even in the order of 1 deg), calculated in time domain, is used tomonitor any small change in the global electrical condition of thecable, because changes in the electrical parameters affect the cableresonance frequencies. Rather than monitoring directly frequencychanges, LIRA monitors the impedance phase, because it can be estimatedin a more reliable and accurate way. This method is suitable forunattended, real-time monitoring of cable global conditions.

FIG. 7 shows an example of impedance phase shift monitoring using theLIRA (Line Resonance Analysis) system. The LIRA system shows in thegraph in the middle left of FIG. 7 the phase shift between a referencesignal CH0 and the resulting signal CH1 modulated by the cable impedanceas a function of time. The phase shift provides information concerningthe insulation degradation.

The reasons why a resonance frequency is used for this purpose are:

-   -   The phase derivative is the highest at resonance, achieving the        maximum sensitivity to cable degradation.    -   The phase changes linearly around a resonance condition. This        makes easier to correlate a phase shift to the insulation        degradation.

Having described preferred embodiments of the invention it will beapparent to those skilled in the art that other embodimentsincorporating the concepts may be used. These and other examples of theinvention illustrated above are intended by way of example only and theactual scope of the invention is to be determined from the followingclaims.

1-13. (canceled)
 14. Method for monitoring a condition of an electricalcable, comprising the steps: calculating a complex cable impedance as afunction of frequency specified by an amplitude and a phase, furthertranslating into a domain t′f→t′  and introducing a radial frequency ω′ in said domain t′$\frac{4\pi \; d}{v_{r}v_{0}}->\omega^{\prime}$  wherein v₀ is thespeed of light in vacuum, and v_(r) is an estimated relative phasevelocity of an electrical signal in the cable; performing a powerspectrum analysis of both amplitude and phase of the cable impedance inthe transformed domain of t′ to find and localize any local degradationto the cable insulation; and identifying frequency components f″₁, f″₂,. . . f″_(n), in the power spectrum of the transformed domain due towave reflections at locations x₁, x₂, . . . , x_(n) along the cable,said wave reflections being due to discontinuities in electricalparameters of said cable, and calculating each of said locations x_(i)by: $x_{i} = {\frac{v_{r}v_{0}f_{i}^{''}}{2}.}$
 15. The method ofclaim 14, characterized in that an estimate of said relative phasevelocity (v_(r)) is provided by: evaluating at least two resonancefrequencies of the cable impedance (Z_(DUT)), identifying twoconsecutive resonance frequency values f_(k) and f_(k+1) respectively,of the cable impedance (Z_(DUT)); calculating a first value of arelative phase velocity v_(r) of the cable by applyingv _(r)=2L(f _(k+1) −f _(k))/v ₀, where L is the length of the cable;calculating a fundamental frequency f′ of said cable using said firstrelative phase velocity v_(r) and applying${f^{\prime} = \frac{2L}{v_{r}v_{0}}},$ calculating a second and moreaccurate value f″ of said fundamental frequency f′ by finding a maximumpeak value in the Fourier transform in the domain of t′ in a selectableinterval around f′; and calculating said estimate of said relative phasevelocity v^(final) _(r) by applying$v_{r}^{final} = \frac{2L}{v_{0}f^{''}}$
 16. The method of claim 14,characterized by a modulator connected to the electrical cable outputs areference signal (CH0), that is a white noise signal, to the cable andto an analyzer, and receives a reflected signal from the cable andoutputs the modulated signal (CH1) to said analyzer, said modulatedsignal (CH1) being the signal which is phase and amplitude modulated bythe cable impedance, and that said analyzer estimates the amplitude andphase of the cable impedance on the basis of the signals (CH0, CH1)received from said modulator.
 17. A system for monitoring a condition ofan electrical cable, comprising generating means for generating amultifrequency wave that is to be modulated by the cable impedance, saidsystem characterized in that it further comprises an analyzer forperforming a power spectrum analysis of both amplitude and phase of thecable impedance in the domain of t′ to find and localize any localdegradation of the cable insulation, and for identifying frequencycomponents f″₁, f″₂, . . . f″_(n) in the power spectrum due to wavereflections at locations x₁, x₂, . . . , x_(n) along the cable, saidwave reflections being due to  discontinuities in electrical parametersof said cable, and for calculating each of said locations x_(i) by:$x_{i} = \frac{v_{r}v_{0}f_{i}^{''}}{2}$  wherein v₀ is the speed oflight in vacuum, and v_(r) is an estimated relative phase velocity of anelectrical signal in the cable.
 18. The system of claim 17,characterized in that said analyzer further is operative to evaluate atleast two resonance frequencies of the cable impedance (Z_(DUT)),identify two consecutive resonance frequency values f_(k) and f_(k+1)respectively, of the cable impedance (Z_(DUT)); calculate a first valueof a relative phase velocity v_(r) of the cable by applyingv _(r)=2L(f _(k+1) −f _(k))/v ₀, where L is the length of the cable;calculate a fundamental frequency f′ of said cable using said firstrelative phase velocity v_(r) and applying${f^{\prime} = \frac{2L}{v_{r}v_{0}}},$ calculate a second and moreaccurate value f″ of said fundamental frequency f′ by finding a maximumpeak value in the Fourier transform in the domain of t′ in a selectableinterval around f′; and finally to calculate said estimate of saidrelative phase velocity v^(final) _(r) by applying$v_{r}^{final} = {\frac{2L}{v_{0}f^{''}}.}$
 19. The system of claim17, characterized by a modulator connected between said generating meansand said analyzer, said modulator also connected to the electrical cablefor outputting a reference signal (CH0) thereto and also to saidanalyzer, and for receiving a reflected signal (CH1) from the cable andoutputting a modulated signal (CH1) to said analyzer, said modulatedsignal (CH1) being phase and amplitude modulated by the cable impedance.